Well I got the thing working with your help. Unfortunately, I don't really understand the network security distinctions so it feels like driving a car by holding on to the rear view mirror.
Here's what I did:
1. Rebooted the router
2. When it restarted I changed the newtork the network name (I suspect that didn't do anything but make me feel like I was doing something useful).
3. I turned security off on the router and was able to use Safari.
4. I used Forget this Network on the iPod
5. Chose WEP security on the router (which it was before)
6. Entered a 10 character alphanumeric string into the router's "WEP Key 1" field
7. Turned the network on in the Touch
8. The iPod then found the network and the the settings looked normal and the little lock icon displayed next to the network name. So I went to Safari and what do you know, it worked.
But what I don't understand is I never got prompted for a password on the Touch or got prompted to Join. It just went directly to selecting the new network name as soon as it was found. I assume it's a secure connection because the little lock icon is there. It could be I did not use Forget the Network the way I recall and perhaps I did things in a somewhat different sequence than I said above.
Regardless, it's working. I want to sincerely thank my two helpers. If it wasn't for you I may have never suspected the router as everything else seemed to be working. I'd probably be throwing the iPod against the wall by now.
Thanks


P.S.
I have a ZyXel X-550 router which most people never heard of. I got it based on Amazon reviews which said it was very easy to set up. It is. This is the first problem I've had with it. Otherwise, it's been extremely reliable, more so that any other I've had. But now I'll be keeping an eye on it.